Photon temporal modes: a complete framework for quantum information science

B. Brecht1?, Dileep V. Reddy2, C. Silberhorn1, and M. G. Raymer2.1Integrated Quantum Optics, Applied Physics, University of Paderborn,

Warburger Strasse 100 33098, Paderborn, Germany and2Oregon Center for Optics, Department of Physics,University of Oregon, Eugene, Oregon 97403, USA

(Dated: October 13, 2015)

Field-orthogonal temporal modes of photonic quantum states provide a new framework for quan-tum information science (QIS). They intrinsically span a high-dimensional Hilbert space and lendthemselves to integration into existing single-mode fiber communication networks. We show thatthe three main requirements to construct a valid framework for QIS – the controlled generation ofresource states, the targeted and highly efficient manipulation of temporal modes and their efficientdetection – can be fulfilled with current technology. We suggest implementations of diverse QISapplications based on this complete set of building blocks.

INTRODUCTION

Quantum information science (QIS) offers means forstoring, transmitting and processing information in waysnot achievable using classical information technology.Examples of the benefits of QIS are unconditionallysecure communication, ultra-precise metrology beyondclassical limits, and superior computational algorithms.

While all of those can theoretically be realized us-ing only photons, it is generally accepted that quan-tum computation will be implemented in material sys-tems, whereas quantum communication and informationtransfer across a distributed quantum network – a so-called “quantum internet” [1] – will be based on photons.Strongly interacting material systems, which can be con-trolled with outstanding precision, facilitate the imple-mentation of stationary logical processors and quantummemories. The latter are an indispensable building blockfor long-distance entanglement distribution via quantumrepeaters, which in turn is inextricably linked with se-cure long-distance quantum communication. Photons,in contrast, interact only weakly with themselves andtheir environment, meaning that they experience verylow decoherence. Thus, they are naturally suited forcarrying fragile quantum information over transmissionlines in a network. The remaining challenge for thesehybrid network architectures is the efficient interfacingof flying qubits (photons) and stationery qubits (ma-terial systems), which is complicated by the fact thatmost practical material systems have stringent require-ments on the photon spectral-temporal amplitude. Thus,small photonic co-processor units that facilitate, for in-stance, the coherent re-shaping of photons in time andfrequency must be available. Note that these do notnecessarily have to fulfill the more stringent demands offault-tolerant quantum computation to be practical andtherefore, as we show, can be realized with current tech-nology.

In this paper we introduce a practical framework forphotonic quantum information science. Our framework

exploits temporal modes (TMs) of single photon states– field-orthogonal broadband wave-packet states – thathave to date not been demonstrated to enable a viablebasis for quantum information encoding. In particular,we complement existing knowledge with all missing build-ing blocks, which are needed to demonstrate that TMssatisfy the three major requirements for the implementa-tion of the photonic subsystems of large-scale quantumnetworks: firstly, for the preparation of good signal carri-ers, appropriate resource states have to be generated andcompletely characterized with high reliability and flex-ibility; secondly, the subsequent processing of quantuminformation in co-processor units requires that controlledoperations can be implemented; finally, efficient detectionschemes, which enable faithful information readout, mustbe available.

We expect that the TM framework for photonic quan-tum information will open avenues towards the realisa-tion of practical QIS applications. One such applicationis the boson sampler [2–6], which, though not on parwith the requirements for fault-tolerant quantum com-putation, may soon show performance beyond the capa-bilities of state-of-the-art classical computer, which arepushed to their limits by linear optical networks withabout 100 modes, of which only 30 are occupied. Our newTM paradigm may offer improved methods to constructlarge networks with reduced switching losses, which arecurrently thought to be the main limiting factor whenconsidering the scalability of photonic quantum process-ing [7].

In the following we first introduce the basic conceptsof our framework by formally defining TMs and their useas an information-encoding basis. Then we briefly reviewthe current state-of-the-art of generating TMs with ultra-fast parametric down-conversion, where we will outlinewhy existing sources do not yet fulfill the requirementsfor QIS with TMs. After this, we highlight recent de-velopments in TM manipulation, which serve as startingpoint for the definition of the complete TM framework.The key enabling findings for this are our recent results,

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which introduce means for sorting TMs with high effi-ciency and selectivity in excess of 99.5 percent. This highefficiency of the “quantum pulse gate” operation can beachieved by dispersion-engineered, multi-stage frequencyconversion driven by spectrally-temporally shaped lasercontrol pulses. We then present new concepts and com-ponents, which enable the establishment of the completeTM framework. In particular, we design the flexible gen-eration of entangled resource states of arbitrary, user-defined dimension, we introduce TM quantum-state to-mography of single-photon as well as photon-pair statesto verify the successful state generation, and we estab-lish concrete applications for QIS. We show that all op-erations necessary to implement photonic co-processorsand quantum communication applications can be imple-mented with TMs. We conclude the paper with a dis-cussion of the experimental challenges and limitations ofour framework.

FUNDAMENTAL CONCEPTS

Starting from a very general point-of-view, we notethat light has four degrees of freedom (DOF), any ofwhich could be used to encode quantum information:these are the helicity and the three components of themomentum vector. In a beam-like geometry these maybe stated as polarization, transverse mode profile (en-compassing two DOFs), and energy (that is, frequency).From these DOFs, polarization is most widely appliedin quantum information processing. The generation ofpolarization-entangled Bell states [8] as resource states isnowadays an established experimental method. Two or-thogonal polarization modes can easily be separated bymeans of using polarizing beamsplitters, and proper gateoperations are readily implemented with linear opticalelements such as waveplates, (polarizing) beamsplittersand detectors. However, polarization intrinsically spansa mere two-dimensional Hilbert space, and thus cannotexploit the true potential of QIS, which, in certain casessuch as quantum key distribution, benefits from higher-dimensional Hilbert spaces [9, 10].

The second DOF, transverse mode profile, has receivedconsiderable attention recently, as it has become appar-ent that the orbital-angular-momentum (OAM) states oflight are a useful basis for encoding information [11–13]and can be efficiently sorted with time-stationary linearoptical elements [14]. They have been used recently todemonstrate, for instance, enhanced security and bitratein quantum communication [15–17]. Still, the OAM basishas three drawbacks limiting its current value for someQIS applications: first, it is inherently incompatible withexisting single-mode fiber networks because informationis encoded onto different spatial field distributions; sec-ond, it is susceptible to medium perturbations such asturbulence, which affects free-space links; and third, the

generation of OAM states with a tailored structure, forinstance a well-defined number of modes, is as of yet anunsolved problem.

Only recently has the final DOF of light – energy, thatis frequency – been recognized as an underutilized re-source for QIS. Because frequency and time are conju-gate variables, we call a set of overlapping but orthog-onal broadband wave-packet modes by the name “tem-poral modes” (TMs). In a coherent-beam-like or single-transverse-mode guided wave geometry, TMs form a com-plete basis for representing an arbitrary state in the en-ergy degree of freedom [18]. TMs overlap in time andfrequency, yet are field-orthogonal. In this respect, theyare analogous to transverse spatial modes, yet they pos-sess distinct advantages. Since all TMs “live” inside thesame spatial field distribution, they are naturally suitedfor use with highly efficient and experimentally robustwaveguide devices and existing single-mode fiber net-works. In addition, they are insensitive to stationary orslowly-varying medium perturbations such as linear dis-persion, due to their overlapping spectra, making themsuitable for real-world applications.

While the TM concept applies to any states of light(e.g. squeezed quadrature states [19, 20]), we restrictourselves to single-photon states to keep this paper con-cise and readable. In this context, TMs are a completemode set for expanding the electromagnetic field and, inaddition, can be regarded as a complete set of quantumstates for single photons.

Temporal modes for single-photon states

For a fixed polarization and transverse field distribu-tion (e.g. in a beam-like geometry), a single-photonquantum state in a specific TM can be expressed as acoherent superposition of a continuum of single-photonstates in different monochromatic modes:

|Aj〉 =

∫dω

2πfj(ω)a†(ω) |0〉 . (1)

Here, a†(ω) is the standard monochromatic creation op-erator and fj(ω) is the complex spectral amplitude ofthe wave packet. By Fourier transform, this same statecan be expressed as a coherent superposition over manypossible “creation times”, and then reads

|Aj〉 =

∫dt fj(t)A

†(t) |0〉 ≡ A†j |0〉 , (2)

where we used the definition

a†(ω) =

∫dt eıωtA†(t); A†(t) =

∫dω

2πe−ıωta†(ω). (3)

In Eq. (2), fj(t) is the temporal shape of the wave packet(defined as the Fourier transform of fj(ω)) and A†(t)

3

FIG. 1. First three members of a TM basis in the frequencydomain (left) and the time domain (right).

creates a photon at time t. We also defined a so-calledbroadband-mode operator

A†j =

∫dt fj(t)A

†(t) =1

2π

∫dω fj(ω)a†(ω), (4)

which creates the wave-packet state |Aj〉. In Fig. 1, weexemplarily plot the first three members of a TM basis,chosen for illustration to be a family of Hermite-Gaussianfunctions of frequency. With this, it is possible to expressevery single-photon temporal wave-packet quantum state|Ψ〉 in a basis of TMs as a superposition of wave-packetstates,

|Ψ〉 =

∞∑j=0

cjA†j |0〉 , (5)

with complex-valued expansion coefficients cj .We want to highlight that, although they fully over-

lap in polarization, space, frequency and time, TMs areorthogonal with respect to a frequency (time) integral

1

2π

∫dω f∗j (ω)fk(ω) =

∫dt f∗j (t)fk(t) = δjk. (6)

They also obey bosonic commutation relations [18, 21]

[Ai, A†j ] = δij (7)

just as do the well-known monochromatic creation oper-ators.

Quantum information encoding with TMs

Deploying TMs for quantum information encoding isan appealing prospect, because TMs span an infinite di-mensional Hilbert space. This has been shown to facil-itate increased information capacity per photon and in-creased security in quantum communication [15–17] when

(a) (b)

FIG. 2. (a) Poincaré sphere. The logical “0” and “1” of a po-larization qubit can be encoded in any two diametrically op-posite points on the sphere. Typically, horizontal and verticalpolarization are deployed. (b) Bloch sphere for TM qubits.Any two orthogonal TMs and their coherent superpositionsmay be used to encode TM qubits. In this example, the TMsare a zeroth and first order Hermite-Gaussian.

compared to two-dimensional encoding. The carriers ofinformation in a d-dimensional Hilbert space are typicallycalled “qudits”.

We define a TM qudit as a coherent superposition ofd TM states:

|ψ〉dTM =

d−1∑j=0

αj |Aj〉 . (8)

To highlight the formal similarity of TMs with otherencoding bases, we start by discussing TM qubits. Themost common implementation of a photonic qubit isthe polarization qubit, which can be written as |ψ〉 =α |H〉 + β |V〉. Here, |H〉 and |V〉 denote horizontal andvertical polarization, respectively, and |α|2 + |β|2 = 1.Commonly, a polarization qubit is represented as a pointon the surface of a Poincaré sphere as sketched in Fig.2(a).

In analogy to this, the definition of a TM qubit requirestwo orthogonal states with which we associate the logical“0” and “1”. Without loss of generality, we can considerzeroth-order and first-order Hermite-Gaussian functionsof frequency to define the TMs, labeled and , andconsequently write

|0〉 ≡ | 〉 , |1〉 ≡ | 〉 . (9)

Then, a TM qubit is given by

|ψ〉TM ≡ α | 〉+ β | 〉 , (10)

where again |α|2 + |β|2 = 1. Similar to polarizationqubits, the TM qubit is best visualized as a point onthe surface of a Bloch sphere as shown in Fig. 2(b).

Mutually unbiased bases

Sets of bases, for which the overlap between a basisvector of one basis with any basis vector from any of the

4

FIG. 3. The columns show the three MUBs for a TM qubit,with the fundamental TM shapes being a zeroth and firstorder Hermite-Gaussian, respectively. The colored areas arethe spectral amplitude, whereas the dark lines are the spectralphases of the TMs, the color-coding corresponds to Fig. 2(b).Note that in this case, the qubit is encoded in the leftmostbasis.

other bases has the same absolute value, are called mutu-ally unbiased bases (MUBs) [22]. They lie at the heart ofQIS applications such as quantum key distribution [23]or quantum state tomography [24]. The physical mean-ing of MUBs is the following: if a certain quantum stateis an eigenstate of one basis then a measurement in anyother MUB yields a uniformly random result yielding noinformation. Using polarization states, the three sets ofStokes vectors denoting horizontal and vertical, diagonaland anti-diagonal as well as left- and right-circular lightform the typically used MUBs.

For the case of the aforementioned TM qubit from Fig.2(b), the basis modes of the three possible MUBs areindicated by the different colors and we explicitly plotthem in Fig. 3. The color coding corresponds to Fig.2(b). If the qubit was given by |ψ〉TM = | 〉, measuringin either the red or green basis results in “0” (upper row)or “1” (lower row) with a probability of 50%.

The challenge for TMs is the implementation of adevice that facilitates a mode-selective measurement,where the phase coherence plays a particularly importantrole. For a polarization qubit, an appropriate combina-tion of wave plates and polarizing beamsplitters read-ily accomplishes the projection onto the respective basissets. For TMs, the situation is more complicated, sincetime-stationary operations are not sufficient for mode-selectivity and so-called quantum pulse gates have to beemployed [25–28]. We return to this point below, wherewe briefly review the solution to the mode-sorting prob-lem.

STATE-OF-THE-ART

In this section, we briefly summarize the current state-of-the-art in generating and manipulating TM states.Typically, the former is realized with parametric down-conversion, whereas the latter can be achieved by deploy-ing TM selective quantum pulse gates.

TM structure of photon pair states

Today, parametric down-conversion (PDC) in opticalwaveguides is the workhorse for the generation of photon-pair and heralded single-photon states. Notably, PDCgenerates quantum states with a rich intrinsic TM struc-ture, when ultrafast pulses are deployed as pump [29].This structure is decoupled from the transverse spatialmode, which is solely determined by the waveguide ge-ometry. It is encoded in the so-called joint spectral am-plitude (JSA) of the PDC f(ωs, ωi), which can be writtenas [30, 31]

f(ωs, ωi) = α(ωs, ωi) · φ(ωs, ωi). (11)

Here, α(ωs, ωi) is the pump-envelope function, which en-compasses energy conservation and the spectrum of thepump pulses, and φ(ωs, ωi) is the phase-matching func-tion, which describes momentum conservation and de-pends on the medium dispersion.

With that, we denote the photon-pair component ofthe generated state

|ψ〉PDC =

∫dωsdωi f(ωs, ωi)a

†(ωs)b†(ωi) |0, 0〉 , (12)

where a†(ωs) and b†(ωi) are standard monochromatic cre-ation operators for signal and idler photons.

A decomposition of the JSA into two sets of uniquelydefined TM basis functions {f (s)(ωs)} and {f (i)(ωi)},which exhibit pairwise correlations such that

f(ωs, ωi) =

∞∑k=0

√λkf

(s)k (ωs)f

(i)k (ωi) (13)

reveals the underlying TM structure of the PDC state[29]. Here, the expansion coefficients are normalized ac-cording to

∑k λk = 1. We graphically show this expan-

sion for a typical, non-engineered PDC in Fig. 4(a).From Eqs. (12) and (13), we obtain

|ψ〉PDC =

∞∑k=0

√λk |Ak, Bk〉 (14)

where we used again the broadband mode operators fromEq. (4). This expression shows that the PDC excites

5

...

(b)

(a)

FIG. 4. (a) Representation of a general PDC process. The leftmost panel shows the JSA f(ωs, ωi), which is the product ofpump envelope function (black solid lines) and the phasematching function (black dashed lines). This function is decomposedinto two sets of TMs {f (s)(ωs)} and {f (i)(ωi)} with weighting coefficients

√λk. In the central part, we plot the first three TM

pairs. The rightmost panel shows the distribution of expansion coefficients√λk. (b) A dispersion-engineered PDC process

excites only one pair of TMs. The JSA does not exhibit any correlations between signal and idler photons. The distribution ofweighting coefficients

√λk consequently exhibits only a single entry greater than zero.

pairs of TM states |Ak〉 and |Bk〉 with a relative weightof√λk.

For the special case of a dispersion-engineered PDCthat excites only a single pair of TMs (see, for instance[32–37]), the state from Eq. (14) reduces to |ψ〉PDC =|A0, B0〉. This situation is shown in Fig. 4(b). In thiscase, by detecting the photon created in one channel,one heralds the single-photon state in the other channelin a known, pure TM. We note, however, that this isnot sufficient for generating resource states for QIS ap-plications. On the one hand, the general PDC state hasan inadequate structure, because the number of TMs inthe state cannot be precisely controlled. On the otherhand, the single-TM state does not constitute an entan-gled resource state, which is a necessary requirement fordifferent QIS applications.

Coherent manipulation of the TM structure ofsingle-photon states

A major requirement for realizing QIS with TMs is thecoherent manipulation of a state in the TM basis. Thiscan be achieved by deploying so-called quantum pulsegates (QPGs) [25, 27, 28, 38]. Note that although we re-strict our discussions to three-wave mixing implementa-tions of QPGs here, all results can be generalized to four-wave mixing. The underlying physical process of a QPGbased on three-wave mixing is dispersion-engineered sum-frequency generation (SFG) inside a nonlinear opticalwaveguide, where one photon from an ultrafast pumppulse and a “red” quantum signal fuse into a “green”converted output photon. Here, red and green describetwo well-separated frequency bands, for instance 1535nm

(red) and 557 nm (green), respectively [26]. An adaptionof this approach for use with continuous-variable quan-tum states has been proposed in [20]. In four-wave mix-ing implementations, two non-degenerate pump pulsesare used, which facilitate smaller frequency shifts of sin-gle photons as compared to using three-wave mixing [39–41].

An ideal QPG that is mode matched to the TMs ofthe source as defined above acts on an arbitrary single-photon input state |ψ〉in of the form Eq. 8 accordingto

|ψ〉out = Q(η)i |ψ〉in (15)

with

Q(η)i = 1− |Ai〉 〈Ai| − |C〉 〈C|

+ cos θi (|Ai〉 〈Ai|+ |C〉 〈C|)+ sin θi (|C〉 〈Ai| − |Ai〉 〈C|) .

(16)

The cosine term preserves either of the two statesof interest, while the sine term “swaps” them with ef-ficiency sin2(θi). The first three terms enforce unitar-ity. This expression is a family of unitary transforma-tions on the single-photon state space comprised of twonon-overlapping subspaces (here, frequency bands): onespanned by the TM states |Aj〉, and a single TM state |C〉occupying the other. It has an elegant interpretation: theQPG acts as a quantum mechanical beamsplitter, whichoperates on TMs instead of polarization or spatial modes.As detailed in [38, 42], the blue pump pulse spectrumα(ω) defines the targeted “red” input TM state |Ai〉 thatis selected and converted to the “green” output state |C〉with an efficiency given by η = sin2(θi). Note that theQPG can also select superpositions of TM states, when

6

(a)

(b)phaseshift

FIG. 5. (a) Schematic of the QPG operation. The shapeof the blue pump pulse selects one TM from the “red” in-put signal and converts it to the “green” output with an effi-ciency of η. All other signal TMs are completely transmitted.The index i labels the addressed input TM. (b) A Mach-Zehnder/Ramsey like configuration of two successive QPGswith an efficiency of 50% each overcomes the time-orderinglimitations of a single QPG and facilitates the selection andconversion of a single TM with an efficiency of 100%.

the pump pulses are shaped accordingly. The parame-ter θi describes the strength of the QPG operation andcan be tuned with the pump pulse energy, although theshapes of the “red” and “green” modes will change slightlyfor different values of θi, due to time-ordering corrections[43–45] (i.e. the input and output TMs are not identical).For genuine QPG operation, θj = 0 for j 6= i; that is, allTMs that are not addressed are completely transmitted.This situation is sketched in Fig. 5(a).

From Eq. (16) we see two things. First, the QPG con-verts any targeted input state |Ai〉 into the same outputstate |C〉. This is important in light of large networkarchitectures, because it facilitates interference betweenformerly orthogonal TM states after the QPG operation.Second, the QPG can also be operated “backwards”. Inthis case, it accepts one single input state |C〉, whichis coherently reshaped to an arbitrary output TM state|Ai〉. This allows the treatment of the |C〉 frequency bandas a buffer, or “processing” state space, and allows one toperform arbitrary linear operations on TM qudits thatreside in the {|Aj〉}-space using combinations of QPGs,as will be shown below.

A measure to quantify the operation fidelity of a QPGis the so-called temporal mode-selectivity [27]

S =sin4(θi)∑∞j=0 sin2(θj)

≤ 1, (17)

which measures the ratio between the squared conversionefficiency of the selected mode and the conversion efficien-cies of all modes. A mode selectivity of 1 characterizesperfect single-TM operation, whereas a mode selectivityof 0 signifies a total absence of modal selectivity.

It has been shown that the single-stage QPG realiza-tion from Fig. 5(a) cannot exceed a mode selectivity ofS = 0.85 due to the effects of time ordering, which leadto a temporal multimode behavior at conversion efficien-cies exceeding 90% [43, 44]. This limitation can be over-come by utilizing a two-stage Mach-Zehnder/Ramsey likesetup of two successive QPGs with an efficiency of 50%each, which are driven by the same pump pulse shape[27, 28]. We sketch this in Fig. 5(b).

In the two-stage QPG a single photon in the target TMwill be converted into an equal superposition of a “green”and a “red” mode by the first stage, and will then be co-herently fully frequency shifted or back-converted in thesecond stage depending on an externally applied phaseshift to the device. The non-target TM components ofthe photon will not participate in the interferometric con-version process due to their vanishingly small per-stageconversion efficiencies, and will effectively transparentlypass through the device. The need for phase coherenceacross the two stages can be met by deriving the twopump pulses from the same master pulse. In a spe-cific configuration [27, 28], this method also eliminatesthe temporal distortion in the shapes of the “red” and“green” modes due to time-ordering effects, which enablesthe cascading of QPGs without the need for inter-QPGcompensatory TM reshaping. Note that the overall oper-ation of the two QPGs is again collectively described byEq. (16) and that we use the simplified sketch from Fig.5(a) for reasons of convenience from here on. Variousoverall efficiency values can now be achieved by tuningthe interferometric phase shift in-between the two stages(Fig. 5(b)) instead of changing the pump power.

In a recent experiment, the implementation of a single-stage QPG with an TM selectivity of 80% at a conversionefficiency of η = 87% when operated at the single-photonlevel has been demonstrated [26].

Note that alternative approaches to TM-selective SFGare studied in [46–48], which forego group-velocitymatching. Although potentially simpler from an experi-mental point of view, these approaches cannot generallyreach high mode selectivities as defined above [43].

COMPLETING THE TOOL KIT FOR A TM QISFRAMEWORK

In this section, we introduce the missing components,which enable our TM framework. In particular, theseare the generation of TM states with an arbitrary, user-defined dimension and their verification using single-photon and photon-pair TM tomography. Thereafter, we

7

(a)

(b)

FIG. 6. (a) When pumping a dispersion engineered PDC with a 1st order Hermite-Gaussian pulse, the resulting JSA (left) hasa negative part signified by the red color. Note that the pump envelope function is again denoted by solid black lines, whereasthe phase-matching function is shown as dashed black lines. A decomposition of this JSA yields exactly two pairs of TMs(center) with similar expansion coefficients (right). Hence, the generated state is a TM Bell state. (b) By further increasingthe order of the Hermite-Gaussian pump, it is possible to successively add TM pairs to the generated state. This state featuresan extremely well-defined dimensionality, although the relative weights of the modes become unbalanced.

show that ideal QPGs can be used to implement linear-optics single- and pair-photons quantum operations.

TM engineering and TM Bell states

Typical QIS applications require at least the faithfulgeneration of Bell states. In the following, we demon-strate how this can be accomplished for TMs by combin-ing in a very natural way a dispersion-engineered PDCwith pulse-shaping techniques, which are well-establishedin the fields of ultrafast optics and coherent control (fora nice review see [49]).

To this end, we consider shaped pump pulses withHermite-Gaussian spectra given by

α(ωs, ωi) =1√

n!√π2nσ

Hn

(∆ω

σ

)exp

[− (∆ω)2

2σ2

].

(18)Here, ∆ω = ωp − ωs − ωi is the frequency mismatch be-tween the pump, signal and idler fields, Hn(x) is a Her-mite polynomial of order n and σ is the spectral 1/e-width of the pump spectral intensity.

Fig. 6(a) shows an engineered PDC that is driven bya 1st order Hermite-Gaussian pump pulse. The JSA de-composes into

f(ωs, ωi) =1√2

(f(s)0 (ωs)f

(i)0 (ωi) + f

(s)1 (ωs)f

(i)1 (ωi)

).

(19)This result can be interpreted such that the PDC com-prises exactly two pairs of TMs with equal excita-tion probability. Consequently, we write the generated

photon-pair state as

|ψ〉PDC ≈1√2

(|A0, B0〉+ |A1, B1〉) =

1√2

(| s, i〉+ | s, i〉) ,(20)

where the graphical representation in the second linehighlights the shapes of the individual signal and idlerTMs. This state is a TM |ψ+〉 Bell state, which is afundamental resource for QIS applications.

In Fig. 6(b), we consider a 2nd order Hermite-Gaussianpump pulse. The decomposition of the resulting JSAshows that the generated state comprises exactly threeTM pairs. Although the relative weights are not evenlydistributed anymore, the dimensionality of the state iswell-defined. Further increasing the order of the pumpHermite-Gaussian pulse successively adds additional TMpairs to the structure of the generated state.

In this way it is possible to generate high-dimensionalphotonic states with an unprecedented degree of control.We emphasize again that all TMs “live” inside the sametransverse spatial waveguide mode, which makes our ap-proach exceptionally robust and guarantees experimentalsimplicity.

Photon TM-state tomography

With the ability to generate TM states with arbitrarydimension, the missing element to render a QIS frame-work based on TMs feasible is the verification of the stategeneration. To this end, we require TM state tomog-raphy, where the challenge is to retrieve the (complex-valued) entries of a quantum state’s density matrix in

8

(a)

(b)

FIG. 7. (a) TM state tomography of a single-photon statewith density matrix ρ. Both transmitted and converted out-put of the QPG are detected with single-photon detectors.(b) Generalized scheme for the TM tomography of a bipho-ton state. Photons “1” and “2” are sent to two different QPGsand the transmitted and converted outputs are detected withsingle photon detectors.

a basis of TMs. This differs from polarization-state to-mography because of the higher dimensionality of theTM-state space. For an arbitrary single-photon state,the density matrix is given by

ρ =∑i,j

Cij |Ai〉 〈Aj | , (21)

with associated TMs {fi(ω)}. This state can be analyzedwith a QPG, which selects a coherent superposition ofTMs given by ζ fk(ω)+

√1− ζ2eıφfl(ω), where ζ ∈ [0, 1],

as shown in Fig. 7(a). This function is defined by theshape of pump pulse the QPG is “programmed” with.Detecting both the converted output and the transmit-ted light with single photon detectors, we measure theaverage converted count rates RC and RT respectively,

which are related to elements of the input density matrixby

RC

RC +RT= ζ2Ckk + (1− ζ2)Cll

+ 2Re[ζ√

1− ζ2eıφClk].

(22)

From this expression we see that for ζ = 0 and ζ = 1,we directly obtain Ckk and Cll, respectively. To retrievethe complex coefficient Clk, we set ζ = 1√

2and evalu-

ate the counts for φ = 0 and φ = π2 . By extension,

we also obtain Ckl and thus a complete subset of ma-trix coefficients of the density matrix ρ. In this way,the complete density matrix or an experimentally feasi-ble subset thereof can be sampled. It is important tonote that any chosen portion of the density matrix canbe “directly” measured in this way without reconstruct-ing the entire state. This is true only for a QPG that canachieve unit selectivity, although without high selectivity,the elements can still be found up to an unknown normal-ization constant. This would necessitate measuring theentire matrix (or making small-magnitude assumptionsabout the unmeasured coefficients).

This procedure can be generalized to certain biphotonstates as sketched in Fig. 7(b). A general two-photonstate in two different spatial modes (with photon labelsA and B) may be expressed in two sets of TM bases as

ρ =∑i,j,k,l

Cijkl |Ai, Bj〉 〈Ak, Bl| . (23)

The two photons are analyzed with two sepa-rate QPGs, which select TMs given by ζA fm(ω) +√

1− ζ2AeıφAfn(ω) and ζB fp(ω)+√

1− ζ2BeıφBfq(ω), re-spectively. Then we employ four single-photon detectorslabeled CA, TA, CB, and TB, as shown in Fig. 7(b).We can then measure coincidence rates between pairs ofdetectors (say between CA and CB, denoted by RCA,CB ,and so on). The following expression of such coincidencerates

RCA,CBRCA,CB +RCA,TB +RTA,CB +RTA,TB

(24)

can be expressed in terms of the biphoton density matrixelements thusly

ζ2Aζ2BCmppm + (1− ζ2A)(1− ζ2B)Cnqqn + ζ2A(1− ζ2B)Cmqqm + (1− ζ2A)ζ2BCnppn

2Re[eıφAζA

√1− ζA

(ζ2BCmppn + (1− ζ2B)Cmqqn

)+ eıφBζB

√1− ζB

(ζ2ACmpqm + (1− ζ2A)Cnqqn

)+ζAζB

√1− ζA

√1− ζB

(eı(φA+φB)Cmpqn + eı(φA−φB)Cmqpn

)] (25)

Cycling through the parameter space (ζ1,2, φ1,2) ∈ {(1,−), (0,−), ( 1√2, 0), ( 1√

2, π2 )} as well as varying the in-

9

PDCherald

PDC herald

(a)

(b)

FIG. 8. (a) Non mode-selective detection of one PDC photongenerally projects its sibling into a mixed state. (b) Deployinga QPG to herald a single TM yields a pure heralded broad-band photon at the cost of a lowered heralding rate.

dices (m,n, o, p) will reveal any desired set of coefficientsfrom the two-photon density matrix.

QIS APPLICATIONS

In this section, we combine the different building blocksto detail several QIS applications, which can be realizedin the TM framework and highlight its versatility. Wefirst consider photon TM purification and TM reshaping,then move on to quantum communication scenarios andconclude with considerations on single-qubit gate opera-tions and cluster-state generation. Note that we will dis-cuss the technical challenges that have to be faced whenimplementing these applications in detail in the followingsection.

Photon TM “purification”

Let us consider an application, which requires eithera photon-pair at very specific wavelengths or a choiceof nonlinear material, such that it is not possible todirectly implement a dispersion-engineered PDC sourcewhich generates only a single pair of TMs, but insteada general PDC state as sketched in Fig. 4(a). In thiscase, people typically resort to spectrally narrow inten-sity filtering to facilitate the heralding of approximatelypure single photons, thus discarding the greater portionof the generated photon pairs [50, 51]. Our TM toolkit provides a more efficient and elegant solution to thisproblem, which additionally facilitates the heralding ofgenuinely pure broadband single photons from a corre-lated source such as shown in Fig. 4(a).

We assume the general PDC state from Eq. (14) anddetect one of the photons, say photon A, with an un-filtered single-photon detector as sketched in Fig. 8(a).This heralds photon B with a reduced density matrixthat is given by

ρB =

∞∑k=0

λk |Bk〉 〈Bk| , (26)

FIG. 9. TM reshaping of a single photon. A QPG first con-verts the “red” single photon to the “green” channel. A secondQPG then reshapes the photon during back-conversion.

which is generally a mixed state with purity P =∑k λ

2k.

On the other hand, we can send photon A to a QPG,which acts as a complex spectral-amplitude shape “filter”that selects a single TM f

(s)i (ωs) with efficiency η, and

detect only the converted output. In this case, a success-ful detection heralds photon B, which is in a pure statewith corresponding density matrix

ρB = |Bi〉 〈Bi| , (27)

as sketched in Fig. 8(b) [25]. Note that this “purifica-tion” comes at the cost of a lower heralding rate, whichis reduced by the factor λi. Still, the advantage is thata photon in a desired TM can be created, rather thansimply a spectrally filtered photon.

As a side remark, although we restricted our analy-sis to photon-pair states, the TM framework can directlybe applied to continuous variable states. In this con-text, a particularly important non-Gaussian operation isTM-selective photon subtraction from a multimode state,which is required for entanglement distillation [52]. It isbased on the same operation as the photon TM “purifi-cation”, but uses a QPG that is intentionally operated atvery low conversion efficiency [20].

Single photon TM reshaping

Large-scale networks require an efficient interfacing be-tween distinct nodes. For different photon sources, thismeans that the photons have to be made indistinguish-able. For coupling photons to solid-state systems, thismeans that the TM of the photons has to match the ac-ceptance TM of the system. In both cases, a coherentTM reshaping of the photons is preferable to other fil-tering operations, since the latter introduce prohibitivelosses. In Fig. 9, we sketch a TM reshaper: A first QPGconverts the “red” input photon – which we implicitlyassume to be pure and thus TM single-mode – to the“green” channel; A second QPG is then used to back-convert the photon to the “red” channel. However, herewe match the shape of the bright pump pulse to the re-quired TM and by this reshape the photon. Note thatthe reshaped mode does not have to be a mode from theoriginal photon TM basis, which is indicated by the label

10

Alice

Bob

Channel 1 Channel 2 Channel 3

Channel 1 Channel 2

Channel 3

FIG. 10. In a TM multiplexing scenario, Alice uses orthog-onal TMs as independent channels, which are sent to Bobin one single physical fiber. He de-multiplexes the channelswith QPGs and reads out the information. The QPGs are be-ing employed as TM multiplexers (Alice) and demultiplexers(Bob) on a single-mode optical channel.

A (as opposed to a numeric label) of the QPG operationin the figure. The complete reshaping operation can thenbe written as

|ψ〉A = Q(1.0)A Q

(1.0)0 |A0〉

= Q(1.0)A |C〉 = − |AA〉 ,

(28)

where we assumed the original photon to be in the TMstate |A0〉 and the overall phase of the output state can beneglected. The operators Q(1.0)

i are the QPG operatorsfrom Eq. (16). In principle, arbitrary reshaping is possi-ble in this way. Note that a reshaping of the “green” TMcan be realized by tailoring the phasematching functionof the QPG [53–55]. In this way, an adapted interfacebetween photons at telecommunication wavelengths andspecific quantum memories can be realized with a singleQPG.

Quantum communication

Another important aspect of QIS is quantum commu-nication (QC), where quantum information is transmit-ted between distant parties, by convention called Aliceand Bob. To this end, information has to be encoded atAlice’s location and decoded and read out at Bob’s lo-cation. Deploying the aforementioned devices and meth-ods, a QC system based on TMs can be readily set up.

Here, we discuss two approaches to realizing this. Thefirst approach utilizes different TMs as different commu-nication channels and thus relies on TM multiplexing.

Note that in this approach, information is not encodedin the TMs but in another degree of freedom, for in-stance the polarization. The second approach directlyencodes the information in arbitrary superpositions ofsingle-photon TMs, and thereby can implement genuinehigh-dimensional QC.

The use of TMs for channel multiplexing would be dis-tinguished from conventional time- or frequency-basedoptical multiplexing, which use either separated shortpulses or narrow spectral windows to define differ-ent information channels. Such schemes have recentlybeen proposed in the general context of QIS as well[56, 57]. However, they are not based on genuinely field-orthogonal modes, which translates to a lower “packingdensity” of signal channels in time-frequency space toensure approximate orthogonality. A fundamental ad-vantage of our TM approach is that it is intrinsicallybased on genuinely field-orthogonal wave-packet modes,which provide in-principle zero cross talk between modechannels, while densely packing these modes in time-frequency phase space.

In QC, for a TM multiplexing a scheme to work,add/drop functionality is essential. Using the QPG, bothoperations can be implemented as sketched in Fig. 10(a).On Alice’s side, a succession of QPGs adds different chan-nels to the communication line. This is possible dueto the TM-selective operation of the QPG, which re-shapes the “green” input f (c)(ω) into the desired “red”TM f

(s)i (ω). At the same time, the existing “red” TMs

f(s)j (ω) with j 6= i are not affected. Note that this oper-ation mode of the QPG has been referred to as quantumpulse shaper earlier [38]. After transmission, Bob deploysa cascade of QPGs to de-multiplex the different channelsinto separate ports, from which the information is readout [46].

The second approach, high-dimensional QC, is appeal-ing in light of quantum key distribution (QKD) applica-tions, where the goal is to establish a secure encryptionkey between Alice and Bob. Deploying TMs, the imple-mentation of a generalized BB84 protocol [23] becomespossible. To clarify this procedure, we first sketch therealization of the original BB84 protocol using two TMMUBs instead of polarization in Fig. 11(a). Alice ran-domly prepares one of the four possible basis states with aQPG and sends it to Bob. Bob in turn randomly choosesthe measurement basis of his QPG and directly detectsboth output ports, which then correspond to ‘0’ and ‘1’.Thereafter, Alice and Bob publicly announce their prepa-ration and measurement bases and keep only those eventswhen both coincide. Sacrificing a part of the so retainedkey, Alice and Bob can uncover an eavesdropper by the25% error he or she inevitably introduces.

This scheme is readily extended to d dimensions. Weillustrate this for the case of d = 4, which is depictedin Fig. 11(b). In this case, five MUBs and thus a total

11

"0""1"

clock

Alice BobEve

random shape selectors

clock

(a)

"0" "1"

Alice BobEve

(b)

"3"

"2"

MUB 1 MUB 2 MUB 3 MUB 4 MUB 5

MUB 1

MUB 2

MUB 3

MUB 4

MUB 5

"0" "1" "3""2"

random basis selection

FIG. 11. (a) Implementation of the BB84 QKD protocol with TMs. Alice randomly prepares one of four possible basis statesand sends it to Bob, who randomly measures in one of two MUBs. The two outputs of Bobs QPG correspond to “0” and “1”.(b) Generalized BB84 in a four-dimensional encoding scheme. Alice randomly prepares one of the 20 possible basis states. Bobchooses randomly one of the five MUBs to measure. Note that in this case he requires three QPGs to fully resolve the fourpossible basis states of each MUB.

of 20 possible basis states exist, from which Alice ran-domly chooses one. The four basis states of each MUBnow encode logical “0” to “3”. In the figure, we used thefirst four Hermite-Gaussian pulses as the “mother” basisfrom which “daughter” MUBs are created. Again, Al-ice transmits the chosen state to Bob who performs thereadout in a randomly chosen basis. Note however thatBob now requires three QPGs to completely separate thefour basis states of the MUBs. More generally, Bob re-quires d − 1 QPGs for a basis of size d. There are twomajor advantages to high-dimensional encoding schemesin QC. On the one hand, high-dimensional encoding fa-cilitates a higher information capacity per photon, andthus leads to a reduction in the overall number of re-quired photons. On the other hand, it has been shownthat high-dimensional encoding can increase the securityof quantum key distribution, due to a larger error that isintroduced by a potential eavesdropper when intercept-ing the transmission [9, 10].

Quantum computation

In this section, we discuss two routes towards quan-tum computation enabled by the completion of the TMtool kit. First, we consider linear optical quantum com-putation (LOQC), where TM qubits propagate througha linear optical network and are subject to single- andtwo-qubit operations, which define the computation algo-rithm. Then, we investigate cluster-state quantum com-putation, where multiple TM qubits are fused in a specificway to create a graph state with a tailored entanglementstructure. Then, measurements of the nodes (photons) ofthe cluster state implement the computation algorithm,the result of which can be read out from the remainingnodes. Although universal photonic quantum computa-tion is beyond today’s technological capabilities [7], therequired operational building blocks can be realized withTMs.

Since in this paper we focus on three-wave mixing im-plementations of QPGs, we are effectively restricted to

12

one single “green” output TM state |C〉, though we allowfor a complete set of “red” input TM states |Ai〉. Con-sequently, the input states are treated as the primaryqudit information “register” space, and the output chan-nel will play the role of a “processing” space. Note thatthis behavior gives rise to the question whether QPGsare sufficient to realize all of the necessary operations forquantum computation. We will show in the followingthat they are.

LOQC

In LOQC, deterministic two-qubit operations are prov-ably impossible. However, arbitrary single qubit opera-tions can be implemented with a combination of QPGs.For this, we require two special cases of the QPG oper-ation from Eq. (16). First, a QPG with a conversionefficiency of 100%, and second, a QPG with a conversionefficiency of 50%. They are represented by operators

Q(1.0)i = 1− |Ai〉 〈Ai| − |C〉 〈C|

+ |C〉 〈Ai| − |Ai〉 〈C| ,(29)

and

Q(0.5)i = 1− |Ai〉 〈Ai| − |C〉 〈C|

+1√2

(|Ai〉 〈Ai|+ |C〉 〈C|)

+1√2

(|C〉 〈Ai| − |Ai〉 〈C|) .

(30)

In Fig. 12, we show how these operations driven by theproper pump shapes can be sequentially combined withchannel-dependent phase shifts, which shift the phaseonly in the “green” processing space and are shown asgreen boxes, to implement the following single-qubit op-erations (up to an overall phase) on the {|A0〉 , |A1〉}space:

(a) Hadamard gate:

H =|A0〉+ |A1〉√

2〈A0|+

|A0〉 − |A1〉√2

〈A1| (31)

(b) Pauli-X gate (type I, II):

X = |A1〉 〈A0|+ |A0〉 〈A1| (32)

(c) Pauli-Y gate (type I, II):

Y = −ı |A1〉 〈A0|+ ı |A0〉 〈A1| (33)

(d) Pauli-Z gate:

Z = |A0〉 〈A0| − |A1〉 〈A1| (34)

(b)

(a)

(c)

(d)

(e)

Hadamard gate

Pauli-X gate (type I)

Pauli-X gate (type II)

Pauli-Y gate (type I)

Pauli-Y gate (type II)

Pauli-Z gate

Phase-shift gate

FIG. 12. Implementation of single-qubit gates for LOQCusing QPGs with 100% conversion efficiency (white boxes),QPGs with 50% conversion efficiency (yellow boxes) andphase shifts of the green |C〉 TM (green boxes). Note thatboth the Pauli-X gate and the Pauli-Y gate have two possi-ble experimental implementations, which differ in the orderin which the red TMs f (s)

0 (ω) and f (s)1 (ω) are addressed.

(e) Phase-shift gate:

φ = |A0〉 〈A0|+ eıφ |A1〉 〈A1| (35)

These realizations rely on only two different pumpshapes, corresponding to the “red” TMs f

(s)0 (ω) and

f(s)1 (ω), which encode the logical “0” and “1”. The phase-shift gate can be simplified, if the phase (φ + π) is im-printed onto one of the two pump pulses. Then, thechannel dependent phase shift can be omitted.

Note that the “green” channel is used only internally,whereas the input and output channels are the “red”TMs. This greatly reduces the challenge of maintain-ing phase relations between different frequency bands.It also eliminates the phase-coherence requirement forpump pulses across different red-channel-to-red-channel

13

a

b

"1"

"2"

FIG. 13. Two TM qubits in spatial beams a and b can befused with two QPGs, which select different “red” TM compo-nents from the qubits and selectively frequency-convert them.Then, the “green” outputs of the QPGs are interfered at a50/50 beamsplitter (blue rectangle) and detected with detec-tors “1” and “2”. For more information, see the text.

single-qubit gates, only requiring it for pump pulses in-ternal to any given single-qubit gate. Additionally, thesequential steps can in principle be fabricated in mono-lithic devices, which promises a compact and robust im-plementation with building blocks that are well-suited tobe used in integrated networks.

We also emphasize that, in a manner similar to [58],any single qudit operation can be realized with a concate-nation of the single qubit operations outlined in this sec-tion. Then, the pump shapes have to be chosen such thatthe single qubit gates operate on every two-dimensionalsubspace of the qudit space successively.

Cluster state quantum computation

Finally, we consider the generation of discrete variablecluster states based on TMs. To efficiently grow clusterstates from a supply of resource Bell pairs, we requireseveral operations. Assuming that we already have astock of linear cluster states which we want to merge intotwo-dimensional cluster states, we need local Hadamardtransformations and projective measurements [59]. Wehave already shown how these can be implemented withTMs. More important is the ability to generate linearcluster states from Bell pairs. In order to do so, we haveto rely on qubit fusion. A general method which facili-tates this for polarization qubits has been introduced byBrowne and Rudolph [60], where it was referred to asType-I fusion. Here, we adapt this scheme to operate onTM qubits as defined in Eq. (10).

Two qubits in spatial beams a and b are sent to twoQPGs as sketched in Fig. 13. The QPGs implementthe operation Q

(1.0)0 on qubit a and Q

(1.0)1 on qubit b,

respectively. This means, that the “red” TMs f (s)0,a(ω)

and f(s)1,b(ω) are converted to the green TMs f (c)(ω)a,b.

The two “green” channels are interfered on a balancedbeamsplitter behind the QPGs to erase any distinguish-

ing information and the beamsplitter output ports aredetected by detectors “1” and “2”. The successful de-tection of a single “green” photon heralds the successfulqubit fusion operation, which can be written in terms ofKraus operators

O1,2 =1√2

(|A0〉b 〈A0|a 〈A0|b ∓ |A1〉b 〈A1|a 〈A1|b

),

(36)where the sign depends on whether detector “1” or “2”fires. The state after a successful fusion is given by

|ψ〉fused =1√2

(|A0〉b ∓ |A1〉a) , (37)

which, as expected, denotes again a qubit state. Notethat the two parts of the fused qubit can be determin-istically combined into a single spatial mode with theadd/drop functionality of the QPG discussed in the con-text of quantum communication.

CHALLENGES

In this section, we detail the challenges one faces whenimplementing QIS applications based on TMs. Whilephotonic quantum-information systems are ideal for serv-ing as intermediary between memory, interaction, anddetection resources, they come with known challenges.Most notably, the absence of any direct photon-photoninteraction limits all-optical quantum information pro-cessing to nondeterministic logic gates [61] or cluster-state measurement schemes [62]. When compared withoptical-polarization or beam-path encoding of quantuminformation, the proposed TM encoding brings additionalchallenges, which need to be overcome in order to takeadvantage of the large in-principle benefits of using TMsfor QIS: their relative immunity from channel dispersionand their compatibility with quantum memories in hy-brid QIS systems, where efficient coupling into and out ofdisparate devices is highly dependent on temporal-modematching which can, in principle, be achieved with TMreshaping.

For this, the limiting factor is the bandwidth ∆νPM

of the phasematching function of the QPG, which deter-mines the minimal bandwidth of the reshaped TM. Forthe QPG presented in [26] the spectrum of the “green”TM had a FWHM of ∆λg = 0.14 nm, corresponding to abandwidth of ∆νg ≈ 135GHz, which equals ∆νPM [38].We can calculate the narrowest possible phasematchingbandwidth of a QPG based on a lithium niobate waveg-uide with uniform periodic poling. The maximum waveg-uide length is limited by the size of commercially lithiumniobate crystals to around Lmax ≈ 90mm. Using thisnumber, the resulting phasematching bandwidth is cal-culated to be ∆νPM ≈ 9.7GHz, which is close to themaximum bandwidth of state-of-the-art quantum mem-ories based on Raman interaction in warm Cs vapor of

14

9.2GHz [63]. In addition, recent results on manipulatingthe phasematching function by manipulating the periodicpoling pattern of waveguides [53–55] hold the promise fora future decrease of the effective phasematching band-width. Thus, deploying QPGs as interfaces between fly-ing and stationary qubits is a realistic vision.

An additional complication when interfacing flying andstationary qudits is the required multimode capability ofthe quantum memory. The Raman memory mentionedabove, for instance, can store only a single TM [64]. How-ever, it has recently been shown that a concatenation ofseveral Raman-type memories can overcome this limita-tion and store high-dimensional TM states [65]. Thisresult is a promising step towards the realization of high-dimensional hybrid quantum networks and facilitates theseamless integration of quantum memories into the TMframework.

A further challenge is the achievable loss budget for aQIS application based on TMs. In this context, we high-light again that all TMs live inside the same spatial modeand thus feature low-loss transmission through standardsingle mode fibers. In addition, waveguide to fiber cou-pling with efficiencies exceeding 92% has been demon-strated [66]. Finally, waveguide propagation losses as lowas 0.016 dB/cm in titanium-indiffused lithium niobatewaveguides can be realized with state-of-the-art technol-ogy [67]. In total, we find a total insertion loss of roughly1.0dB for coupling from a fiber to a 90mm long QPGand back to a fiber. In this case, the main losses arisefrom the fiber couplings. We note, that this challengeis not singular to the TM framework, but rather a chal-lenge that the whole field of integrated quantum photon-ics has to face. Although current loss numbers are stillprohibitively high, a significant increase in waveguide tofiber coupling efficiency can be expected in the comingyears, which will alleviate this situation.

Let us now focus on the realistically achievable num-ber of TMs and thus the dimensionality of the accessibleHilbert space. With increasing mode order, the com-plexity as well as the spectral extent of TMs increases.Hence, the number of modes will be bounded, on the onehand, by the resolution of the deployed pulse shapers forpump pulses and, on the other hand, by the maximumspectral bandwidth of single-TM operation of the QPGs.For the demonstrator from [26], the maximum spectralbandwidth can be calculated to be around 25 nm for aninput signal centered around 1550 nm. For larger band-widths, the group-velocity dispersion inside the waveg-uide becomes non-negligible and the process is not TMsingle-mode anymore. Let us then assume that the TMstates are generated with a PDC in a KTP waveguideas presented in [36, 37]. Then, the FWHM of the funda-mental TM is around 5.0 nm. In this case, 10 TMs can beaddressed with a selectivity in excess of 95%. A simpleoptimization of the PDC bandwidth and the length of theQPG waveguide increases this number to 20 TMs. Note

that this is the limit only of the particular realization ofa QPG based on lithium niobate waveguides. Investigat-ing other materials with a more favorable group-velocitydispersion behavior can yield an even higher mode num-ber.

Concerning the resolution of state-of-the-art pulseshapers, we note that spatial light modulators with upto 4096 pixels are commercially available. Paired withproper imaging optics, these devices are capable of shap-ing TMs of order 100 with a fidelity of more than 99.9%.With respect to spatial light modulators, we also notetheir current limited switching speeds, which are typi-cally in the order of few tens of kHz. These impose anupper limit on the switching speed of QIS applications.Again, this challenge does not only affect the TM frame-work, but also QIS based on transverse spatial modes,which also relies on SLMs as key elements.

Now we consider the fidelity of the LOQG gate op-erations. In [28], Reddy et al. investigate the mode-selectivity of two-stage and multi-stage approaches to re-alizing QPGs. They found that in a two-stage QPG, themaximum selectivity is S ≈ 98.46%, which translates toa maximum gate fidelity of around 95.4%, since everyLOQC gate consists of three QPGs. This value cannotcompete with requirements on fault-tolerant LOQC, butmay facilitate small co-processing operations with onlyfew gates. In addition, by increasing the number of stagesin the QPG, the selectivity asymptotically approachesone. Thus there is a trade-off between the TM-selectivityand the total internal losses of a gate operation, whichhas to be evaluated in light of specific applications’ re-quirements.

Finally, let us turn our attention to the synchroniza-tion of the time-dependent, active components driven byshaped laser pulses in a TM framework. The timingrequirements may be more severe when using TMs in-stead of other encoding bases, because the TM schemerelies essentially on temporal orthogonality, which is de-graded under time jitter. To overcome this timing chal-lenge over long-distance transmissions, we envision theuse of weak coherent ‘pilot’ pulses, which when ampli-fied at the receiver can serve as a timing reference, apump pulse, and a transmission-medium induced linear-dispersion compensator, all in one.

In general, we find that, as with all burgeoning frame-works for optical QIS, the use of TMs will require signifi-cant investments in integrated device fabrication technol-ogy and timing electronics. TMs also share with otherframeworks the need for efficient single-photon detec-tion and lossless programmable optical routing. Ulti-mately, TM-based schemes might have to rely on perfor-mance gains from single-mode networkability and higherdimensionality, supplemented by their accommodation ofbroadband quantum memories, to outperform other op-tical QIS frameworks.

15

CONCLUSION

We have shown that TMs of single-photon states forman appealing framework for QIS. Formally, they are com-parable with transverse spatial field modes, but havedistinct advantages over spatial modes: they are nat-urally compatible with waveguide technology, makingthem ideal candidates for integration into existing com-munication networks, and they are not affected by typicalmedium distortions such as linear dispersion, which ren-ders them robust basis states for real-world applications.Still, TMs are of yet an underused resource for QIS.

In this paper, we demonstrated that QIS based on TMsis feasible with current technology. We introduced a novelmethod for the generation of photon pair states compris-ing a user-defined number of TMs, which facilitates inparticular the generation of TM Bell states. This methodrelies on the combination of dispersion-engineered PDCwith classical pulse shaping for the pump pulses of theprocess. We then proposed TM tomography of singlephoton and photon-pair states as building blocks for aQIS framework based on TMs.

Having established the necessary basis, we moved onto the implementation of QIS applications. With smallphotonic co-processing units embedded into large-scalehybrid quantum networks in mind, we first focussed onTM “purification” and TM reshaping. Thereafter, we dis-cussed quantum communication based on TMs, where wepresented two approaches: a TM multiplexing approach,where different TMs represented independent channels,and a high-dimensional TM QKD scenario, where theinformation was encoded into the order of the TMs. Fi-nally, we demonstrated that any single qudit operationcan be implemented with a succession of properly ad-justed QPGs. We concluded the applications section witha scheme for TM cluster state generation which highlightsthe versatility of the TM framework.

Finally, we discussed in detail technical challenges thatmust be faced when implementing QIS based on TMs.We expect that the introduction of this new frameworkwill open novel research avenues in both fundamental andapplied QIS.

DVR and MGR were supported by the NationalScience Foundation through ENG-EPMD and PHYS-QIS. BB and CS acknowledge financial support by theDeutsche Forschungsgemeinschaft (DFG) via Sonder-forschungsbereich TRR 142.

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